Construction of Secure Elliptic Cryptosystems Using CM Tests and Liftings
نویسندگان
چکیده
Elliptic curves over number elds with CM can be used to design non-isogenous elliptic cryptosystems over nite elds e ciently. The existing algorithm to build such CM curves, so-called the CM eld algorithm, is based on analytic expansion of modular functions, costing computations of O(2 5h=2 h 21=4 ) where h is the class number of the endomorphism ring of the CM curve. Thus it is e ective only in the small class number cases. This paper presents polynomial time algorithms in h to build CM elliptic curves over number elds. In the rst part, probabilistic probabilistic algorithms of CM tests are presented to nd elliptic curves with CM without restriction on class numbers. In the second part, we show how to construct ring class elds from ray class elds. Finally, a deterministic algorithm for lifting the ring class equations from small nite elds thus construct CM curves is presented. Its complexity is shown as O(h 7 ).
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